Final answer:
The line that passes through P(4,5) and is perpendicular to the line y = -1/3x - 6 is y = 3x - 7. This is found by taking the negative reciprocal of the original slope and using the point-slope form to obtain the equation.
Step-by-step explanation:
The question asks for an equation of a line that passes through a given point, P(4,5), and is perpendicular to a given line whose equation is y = -1/3x - 6. To find a line perpendicular to another, we must know that the slopes of perpendicular lines are negative reciprocals of each other. Thus, if the slope of the given line is -1/3, the slope of the perpendicular line will be the negative reciprocal, which is 3. We can then use point-slope form to construct our equation, which is y - y1 = m(x - x1), where (x1, y1) is the point the line passes through and m is the slope of the line. Substituting in P(4,5) and the perpendicular slope, we get:
y - 5 = 3(x - 4)
If we want to express this in slope-intercept form, or y = mx + b, we simplify the equation as follows:
y - 5 = 3x - 12
y = 3x - 7
The equation y = 3x - 7 is the line that passes through point P(4,5) and is perpendicular to the line with the equation y = -1/3x - 6.